Step |
Hyp |
Ref |
Expression |
1 |
|
hypcgr.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
hypcgr.m |
⊢ − = ( dist ‘ 𝐺 ) |
3 |
|
hypcgr.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
4 |
|
hypcgr.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
5 |
|
hypcgr.h |
⊢ ( 𝜑 → 𝐺 DimTarskiG≥ 2 ) |
6 |
|
hypcgr.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
7 |
|
hypcgr.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
8 |
|
hypcgr.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
9 |
|
hypcgr.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑃 ) |
10 |
|
hypcgr.e |
⊢ ( 𝜑 → 𝐸 ∈ 𝑃 ) |
11 |
|
hypcgr.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝑃 ) |
12 |
|
hypcgr.1 |
⊢ ( 𝜑 → 〈“ 𝐴 𝐵 𝐶 ”〉 ∈ ( ∟G ‘ 𝐺 ) ) |
13 |
|
hypcgr.2 |
⊢ ( 𝜑 → 〈“ 𝐷 𝐸 𝐹 ”〉 ∈ ( ∟G ‘ 𝐺 ) ) |
14 |
|
hypcgr.3 |
⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐷 − 𝐸 ) ) |
15 |
|
hypcgr.4 |
⊢ ( 𝜑 → ( 𝐵 − 𝐶 ) = ( 𝐸 − 𝐹 ) ) |
16 |
|
hypcgrlem2.b |
⊢ ( 𝜑 → 𝐵 = 𝐸 ) |
17 |
|
hypcgrlem1.s |
⊢ 𝑆 = ( ( lInvG ‘ 𝐺 ) ‘ ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ( LineG ‘ 𝐺 ) 𝐵 ) ) |
18 |
|
hypcgrlem1.a |
⊢ ( 𝜑 → 𝐶 = 𝐹 ) |
19 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) = 𝐵 ) → 𝐺 ∈ TarskiG ) |
20 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) = 𝐵 ) → 𝐶 ∈ 𝑃 ) |
21 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) = 𝐵 ) → 𝐴 ∈ 𝑃 ) |
22 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) = 𝐵 ) → 𝐹 ∈ 𝑃 ) |
23 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) = 𝐵 ) → 𝐷 ∈ 𝑃 ) |
24 |
|
eqid |
⊢ ( LineG ‘ 𝐺 ) = ( LineG ‘ 𝐺 ) |
25 |
|
eqid |
⊢ ( pInvG ‘ 𝐺 ) = ( pInvG ‘ 𝐺 ) |
26 |
1 2 3 24 25 4 6 7 8 12
|
ragcom |
⊢ ( 𝜑 → 〈“ 𝐶 𝐵 𝐴 ”〉 ∈ ( ∟G ‘ 𝐺 ) ) |
27 |
1 2 3 24 25 4 8 7 6
|
israg |
⊢ ( 𝜑 → ( 〈“ 𝐶 𝐵 𝐴 ”〉 ∈ ( ∟G ‘ 𝐺 ) ↔ ( 𝐶 − 𝐴 ) = ( 𝐶 − ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) ‘ 𝐴 ) ) ) ) |
28 |
26 27
|
mpbid |
⊢ ( 𝜑 → ( 𝐶 − 𝐴 ) = ( 𝐶 − ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) ‘ 𝐴 ) ) ) |
29 |
28
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) = 𝐵 ) → ( 𝐶 − 𝐴 ) = ( 𝐶 − ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) ‘ 𝐴 ) ) ) |
30 |
18
|
eqcomd |
⊢ ( 𝜑 → 𝐹 = 𝐶 ) |
31 |
30
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) = 𝐵 ) → 𝐹 = 𝐶 ) |
32 |
1 2 3 4 5 6 9 25 7
|
ismidb |
⊢ ( 𝜑 → ( 𝐷 = ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) ‘ 𝐴 ) ↔ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) = 𝐵 ) ) |
33 |
32
|
biimpar |
⊢ ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) = 𝐵 ) → 𝐷 = ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) ‘ 𝐴 ) ) |
34 |
31 33
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) = 𝐵 ) → ( 𝐹 − 𝐷 ) = ( 𝐶 − ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) ‘ 𝐴 ) ) ) |
35 |
29 34
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) = 𝐵 ) → ( 𝐶 − 𝐴 ) = ( 𝐹 − 𝐷 ) ) |
36 |
1 2 3 19 20 21 22 23 35
|
tgcgrcomlr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) = 𝐵 ) → ( 𝐴 − 𝐶 ) = ( 𝐷 − 𝐹 ) ) |
37 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 = 𝐷 ) → 𝐴 = 𝐷 ) |
38 |
18
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 = 𝐷 ) → 𝐶 = 𝐹 ) |
39 |
37 38
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 = 𝐷 ) → ( 𝐴 − 𝐶 ) = ( 𝐷 − 𝐹 ) ) |
40 |
12
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 〈“ 𝐴 𝐵 𝐶 ”〉 ∈ ( ∟G ‘ 𝐺 ) ) |
41 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 𝐺 ∈ TarskiG ) |
42 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 𝐴 ∈ 𝑃 ) |
43 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 𝐵 ∈ 𝑃 ) |
44 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 𝐶 ∈ 𝑃 ) |
45 |
1 2 3 24 25 41 42 43 44
|
israg |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 〈“ 𝐴 𝐵 𝐶 ”〉 ∈ ( ∟G ‘ 𝐺 ) ↔ ( 𝐴 − 𝐶 ) = ( 𝐴 − ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) ‘ 𝐶 ) ) ) ) |
46 |
40 45
|
mpbid |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐴 − 𝐶 ) = ( 𝐴 − ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) ‘ 𝐶 ) ) ) |
47 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 𝐺 DimTarskiG≥ 2 ) |
48 |
9
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 𝐷 ∈ 𝑃 ) |
49 |
1 2 3 41 47 42 48
|
midcl |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ∈ 𝑃 ) |
50 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) |
51 |
1 3 24 41 49 43 50
|
tgelrnln |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ( LineG ‘ 𝐺 ) 𝐵 ) ∈ ran ( LineG ‘ 𝐺 ) ) |
52 |
|
eqid |
⊢ ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) = ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) |
53 |
|
eqid |
⊢ ( cgrG ‘ 𝐺 ) = ( cgrG ‘ 𝐺 ) |
54 |
1 2 3 24 25 41 43 52 44
|
mircl |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) ‘ 𝐶 ) ∈ 𝑃 ) |
55 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 𝐴 ≠ 𝐷 ) |
56 |
1 2 3 41 47 42 48
|
midbtwn |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ∈ ( 𝐴 𝐼 𝐷 ) ) |
57 |
1 24 3 41 42 49 48 56
|
btwncolg3 |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐷 ∈ ( 𝐴 ( LineG ‘ 𝐺 ) ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) ∨ 𝐴 = ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) ) |
58 |
|
eqidd |
⊢ ( 𝜑 → 𝐷 = 𝐷 ) |
59 |
58 16 18
|
s3eqd |
⊢ ( 𝜑 → 〈“ 𝐷 𝐵 𝐶 ”〉 = 〈“ 𝐷 𝐸 𝐹 ”〉 ) |
60 |
59
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 〈“ 𝐷 𝐵 𝐶 ”〉 = 〈“ 𝐷 𝐸 𝐹 ”〉 ) |
61 |
13
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 〈“ 𝐷 𝐸 𝐹 ”〉 ∈ ( ∟G ‘ 𝐺 ) ) |
62 |
60 61
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 〈“ 𝐷 𝐵 𝐶 ”〉 ∈ ( ∟G ‘ 𝐺 ) ) |
63 |
1 2 3 24 25 41 48 43 44
|
israg |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 〈“ 𝐷 𝐵 𝐶 ”〉 ∈ ( ∟G ‘ 𝐺 ) ↔ ( 𝐷 − 𝐶 ) = ( 𝐷 − ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) ‘ 𝐶 ) ) ) ) |
64 |
62 63
|
mpbid |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐷 − 𝐶 ) = ( 𝐷 − ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) ‘ 𝐶 ) ) ) |
65 |
1 24 3 41 42 48 49 53 44 54 2 55 57 46 64
|
lncgr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) − 𝐶 ) = ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) − ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) ‘ 𝐶 ) ) ) |
66 |
1 2 3 24 25 41 49 43 44
|
israg |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 〈“ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) 𝐵 𝐶 ”〉 ∈ ( ∟G ‘ 𝐺 ) ↔ ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) − 𝐶 ) = ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) − ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) ‘ 𝐶 ) ) ) ) |
67 |
65 66
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 〈“ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) 𝐵 𝐶 ”〉 ∈ ( ∟G ‘ 𝐺 ) ) |
68 |
1 3 24 41 49 43 50
|
tglinerflx1 |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ∈ ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ( LineG ‘ 𝐺 ) 𝐵 ) ) |
69 |
1 3 24 41 49 43 50
|
tglinerflx2 |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 𝐵 ∈ ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ( LineG ‘ 𝐺 ) 𝐵 ) ) |
70 |
1 2 3 41 47 17 24 51 49 52 67 68 69 44 50
|
lmimid |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝑆 ‘ 𝐶 ) = ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) ‘ 𝐶 ) ) |
71 |
70
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐴 − ( 𝑆 ‘ 𝐶 ) ) = ( 𝐴 − ( ( ( pInvG ‘ 𝐺 ) ‘ 𝐵 ) ‘ 𝐶 ) ) ) |
72 |
46 71
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐴 − 𝐶 ) = ( 𝐴 − ( 𝑆 ‘ 𝐶 ) ) ) |
73 |
1 2 3 41 47 48 42
|
midcom |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐷 ( midG ‘ 𝐺 ) 𝐴 ) = ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) |
74 |
73 68
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐷 ( midG ‘ 𝐺 ) 𝐴 ) ∈ ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ( LineG ‘ 𝐺 ) 𝐵 ) ) |
75 |
55
|
necomd |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 𝐷 ≠ 𝐴 ) |
76 |
1 3 24 41 48 42 75
|
tgelrnln |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐷 ( LineG ‘ 𝐺 ) 𝐴 ) ∈ ran ( LineG ‘ 𝐺 ) ) |
77 |
1 2 3 41 42 49 48 56
|
tgbtwncom |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ∈ ( 𝐷 𝐼 𝐴 ) ) |
78 |
1 3 24 41 48 42 49 75 77
|
btwnlng1 |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ∈ ( 𝐷 ( LineG ‘ 𝐺 ) 𝐴 ) ) |
79 |
68 78
|
elind |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ∈ ( ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ( LineG ‘ 𝐺 ) 𝐵 ) ∩ ( 𝐷 ( LineG ‘ 𝐺 ) 𝐴 ) ) ) |
80 |
1 3 24 41 48 42 75
|
tglinerflx2 |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 𝐴 ∈ ( 𝐷 ( LineG ‘ 𝐺 ) 𝐴 ) ) |
81 |
50
|
necomd |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 𝐵 ≠ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) |
82 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 = ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) → 𝐺 ∈ TarskiG ) |
83 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 = ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) → 𝐴 ∈ 𝑃 ) |
84 |
9
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 = ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) → 𝐷 ∈ 𝑃 ) |
85 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 = ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) → 𝐺 DimTarskiG≥ 2 ) |
86 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 = ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) → 𝐴 = ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) |
87 |
86
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 = ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) → ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) = 𝐴 ) |
88 |
1 2 3 82 85 83 84 87
|
midcgr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 = ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) → ( 𝐴 − 𝐴 ) = ( 𝐴 − 𝐷 ) ) |
89 |
88
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 = ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) → ( 𝐴 − 𝐷 ) = ( 𝐴 − 𝐴 ) ) |
90 |
1 2 3 82 83 84 83 89
|
axtgcgrid |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 = ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) → 𝐴 = 𝐷 ) |
91 |
90
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) → ( 𝐴 = ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) → 𝐴 = 𝐷 ) ) |
92 |
91
|
necon3d |
⊢ ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) → ( 𝐴 ≠ 𝐷 → 𝐴 ≠ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) ) |
93 |
92
|
imp |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 𝐴 ≠ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) |
94 |
1 2 3 4 6 7 9 10 14
|
tgcgrcomlr |
⊢ ( 𝜑 → ( 𝐵 − 𝐴 ) = ( 𝐸 − 𝐷 ) ) |
95 |
16
|
oveq1d |
⊢ ( 𝜑 → ( 𝐵 − 𝐷 ) = ( 𝐸 − 𝐷 ) ) |
96 |
94 95
|
eqtr4d |
⊢ ( 𝜑 → ( 𝐵 − 𝐴 ) = ( 𝐵 − 𝐷 ) ) |
97 |
96
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐵 − 𝐴 ) = ( 𝐵 − 𝐷 ) ) |
98 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) = ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) |
99 |
1 2 3 41 47 42 48 25 49
|
ismidb |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐷 = ( ( ( pInvG ‘ 𝐺 ) ‘ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) ‘ 𝐴 ) ↔ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) = ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) ) |
100 |
98 99
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 𝐷 = ( ( ( pInvG ‘ 𝐺 ) ‘ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) ‘ 𝐴 ) ) |
101 |
100
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐵 − 𝐷 ) = ( 𝐵 − ( ( ( pInvG ‘ 𝐺 ) ‘ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) ‘ 𝐴 ) ) ) |
102 |
97 101
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐵 − 𝐴 ) = ( 𝐵 − ( ( ( pInvG ‘ 𝐺 ) ‘ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) ‘ 𝐴 ) ) ) |
103 |
1 2 3 24 25 41 43 49 42
|
israg |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 〈“ 𝐵 ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) 𝐴 ”〉 ∈ ( ∟G ‘ 𝐺 ) ↔ ( 𝐵 − 𝐴 ) = ( 𝐵 − ( ( ( pInvG ‘ 𝐺 ) ‘ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ) ‘ 𝐴 ) ) ) ) |
104 |
102 103
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 〈“ 𝐵 ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) 𝐴 ”〉 ∈ ( ∟G ‘ 𝐺 ) ) |
105 |
1 2 3 24 41 51 76 79 69 80 81 93 104
|
ragperp |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ( LineG ‘ 𝐺 ) 𝐵 ) ( ⟂G ‘ 𝐺 ) ( 𝐷 ( LineG ‘ 𝐺 ) 𝐴 ) ) |
106 |
105
|
orcd |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ( LineG ‘ 𝐺 ) 𝐵 ) ( ⟂G ‘ 𝐺 ) ( 𝐷 ( LineG ‘ 𝐺 ) 𝐴 ) ∨ 𝐷 = 𝐴 ) ) |
107 |
1 2 3 41 47 17 24 51 48 42
|
islmib |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐴 = ( 𝑆 ‘ 𝐷 ) ↔ ( ( 𝐷 ( midG ‘ 𝐺 ) 𝐴 ) ∈ ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ( LineG ‘ 𝐺 ) 𝐵 ) ∧ ( ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ( LineG ‘ 𝐺 ) 𝐵 ) ( ⟂G ‘ 𝐺 ) ( 𝐷 ( LineG ‘ 𝐺 ) 𝐴 ) ∨ 𝐷 = 𝐴 ) ) ) ) |
108 |
74 106 107
|
mpbir2and |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → 𝐴 = ( 𝑆 ‘ 𝐷 ) ) |
109 |
108
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐴 − ( 𝑆 ‘ 𝐶 ) ) = ( ( 𝑆 ‘ 𝐷 ) − ( 𝑆 ‘ 𝐶 ) ) ) |
110 |
1 2 3 41 47 17 24 51 48 44
|
lmiiso |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( ( 𝑆 ‘ 𝐷 ) − ( 𝑆 ‘ 𝐶 ) ) = ( 𝐷 − 𝐶 ) ) |
111 |
18
|
oveq2d |
⊢ ( 𝜑 → ( 𝐷 − 𝐶 ) = ( 𝐷 − 𝐹 ) ) |
112 |
111
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐷 − 𝐶 ) = ( 𝐷 − 𝐹 ) ) |
113 |
109 110 112
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐴 − ( 𝑆 ‘ 𝐶 ) ) = ( 𝐷 − 𝐹 ) ) |
114 |
72 113
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) ∧ 𝐴 ≠ 𝐷 ) → ( 𝐴 − 𝐶 ) = ( 𝐷 − 𝐹 ) ) |
115 |
39 114
|
pm2.61dane |
⊢ ( ( 𝜑 ∧ ( 𝐴 ( midG ‘ 𝐺 ) 𝐷 ) ≠ 𝐵 ) → ( 𝐴 − 𝐶 ) = ( 𝐷 − 𝐹 ) ) |
116 |
36 115
|
pm2.61dane |
⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) = ( 𝐷 − 𝐹 ) ) |